Numerical Simulation of Biofilm Formation in a Microchannel

  • David Landa-MarbánEmail author
  • Iuliu Sorin Pop
  • Kundan Kumar
  • Florin A. Radu
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)


The focus of this paper is the numerical solution of a mathematical model for the growth of a permeable biofilm in a microchannel. The model includes water flux inside the biofilm, different biofilm components, and shear stress on the biofilm-water interface. To solve the resulting highly coupled system of model equations, we propose a splitting algorithm. The Arbitrary Lagrangian Eulerian (ALE) method is used to track the biofilm-water interface. Numerical simulations are performed using physical parameters from the existing literature. Our computations show the effect of biofilm permeability on the nutrient transport and on its growth.



The work of DLM and FAR was partially supported by the Research Council of Norway through the projects IMMENS no. 255426 and CHI no. 255510. ISP was supported by the Research Foundation-Flanders (FWO) through the Odysseus programme (G0G1316N) and Statoil through the Akademia grant.


  1. 1.
    E. Alpkvist, I. Klapper, A multidimensional multispecies continuum model for heterogeneous biofilm development. Bull. Math. Biol. 69, 765–789 (2007)CrossRefGoogle Scholar
  2. 2.
    G. Beavers, D. Joseph, Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30(1), 197–207 (1967)CrossRefGoogle Scholar
  3. 3.
    W. Deng et al., Effect of permeable biofilm on micro-and macro-scale flow and transport in bioclogged pores. Environ. Sci. Technol. 47(19), 11092–11098 (2013)CrossRefGoogle Scholar
  4. 4.
    J. Donea et al., Arbitrary Lagrangian–Eulerian methods. Encycl. Comput. Mech. 1(14), 413–437 (2004)Google Scholar
  5. 5.
    R.M. Donlan, Biofilms: microbial life on surfaces. Emerg. Infect. Dis. 8(9), 881–890 (2002)CrossRefGoogle Scholar
  6. 6.
    R. Duddu, D.L. Chopp, B. Moran, A two-dimensional continuum model of biofilm growth incorporating fluid flow and shear stress based detachment. Biotechnol. Bioeng. 103, 92–104 (2009)CrossRefGoogle Scholar
  7. 7.
    H.C. Flemming, J. Wingender, The biofilm matrix. Nat. Rev. Microbiol. 8(9), 623–633 (2010)CrossRefGoogle Scholar
  8. 8.
    D. Landa-Marbán et al., A pore-scale model for permeable biofilm: numerical simulations and laboratory experiments (2018, under review)Google Scholar
  9. 9.
    F. List, F.A. Radu, A study on iterative methods for solving Richards’ equation. Comput. Geosci. 20(341), 341–353 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. Mikelic, W.Jäger, On the interface condition of Beavers, Joseph, and Saffman. SIAM J. Appl. Math. 60(4), 1111–1127 (2000)MathSciNetCrossRefGoogle Scholar
  11. 11.
    I.S. Pop, F. Radu, P. Knabner, Mixed finite elements for the Richards equation: linearization procedure. J. Comput. Appl. Math. 168(1–2), 365–373 (2004)MathSciNetCrossRefGoogle Scholar
  12. 12.
    F.A. Radu et al., A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media. J. Comput. Appl. Math. 289, 134–141 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    P.G. Saffman, On the boundary condition at the surface of a porous medium. Stud. Appl. Math. 50(2), 93–101 (1971)CrossRefGoogle Scholar
  14. 14.
    R. Schulz, P. Knabner, Derivation and analysis of an effective model for biofilm growth in evolving porous media. Math. Meth. Appl. Sci. 40, 2930–2948 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    T.L. van Noorden et al., An upscaled model for biofilm growth in a thin strip. Water Resour. Res. 46, W06505 (2010)Google Scholar
  16. 16.
    B. Vu et al., Bacterial extracellular polysaccharides involved in biofilm formation. Molecules 14(7), 2535–2554 (2009)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • David Landa-Marbán
    • 1
    Email author
  • Iuliu Sorin Pop
    • 1
    • 2
  • Kundan Kumar
    • 1
  • Florin A. Radu
    • 1
  1. 1.University of BergenBergenNorway
  2. 2.Faculty of SciencesHasselt UniversityDiepenbeekBelgium

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